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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas for vertical intervals > For Re(z0/2 Pi-3/4) ∈ Z





http://functions.wolfram.com/08.05.06.0031.01









  


  










Input Form





EllipticF[z, m] == (2 (2 Re[Subscript[z, 0]/(2 Pi) - 3/4] + 1) EllipticK[m] + (1/Sqrt[m]) EllipticK[1/m]) (1 - Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])]) + Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])] EllipticF[Subscript[z, 0], m] + Sum[(1/k!) Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[Subscript[z, 0]]^q (2 p + q - j)^k Binomial[j - q, p] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/( (j - i - 1)! (2 Sin[Subscript[z, 0]])^(j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/2, s] Pochhammer[1/2, i - s] m^(i - s) Cos[Subscript[z, 0]]^(-1 - 2 s) (1 - m Sin[Subscript[z, 0]]^2)^(-(1/2) - i + s), {s, 0, i}], {i, 0, j - 1}])/E^((1/2) I ((k - 2 p - q + j) Pi + 2 (2 p + q - j) Subscript[z, 0])), {p, 0, j - q}], {q, 0, j - 1}], {j, 1, k}] (z - Subscript[z, 0])^k, {k, 1, Infinity}] /; Element[Re[Subscript[z, 0]/(2 Pi) - 3/4], Integers]










Standard Form





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MathML Form







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Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02





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